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Question Number 163496 by Ahmed777hamouda last updated on 07/Jan/22

$${\int }_0^{\frac{\pi }{2}}256{\mathit{c}\mathit{o}\mathit{s}}^5\left(\frac{\mathit{x}}{2}\right){\mathit{s}\mathit{i}\mathit{n}}^{11}\left(\frac{\mathit{x}}{2}\right)\mathit{d}\mathit{x}$$

Answered by Ar Brandon last updated on 07/Jan/22

$$I={\int }_0^{\frac{\pi }{2}}{256\cos }^5\left(\frac{x}{2}\right){\sin }^{11}\left(\frac{x}{2}\right)dx=8{\int }_0^{\frac{\pi }{2}}{\sin }^{5}x{\left(\frac{1-\cos x}{2}\right)}^{3}dx$$$$={\int }_0^{\frac{\pi }{2}}{\sin }^{5}x(1-3\cos x+{3\cos }^{2}x-{\cos }^{3}x)dx$$$$=\frac{1}{2}\beta (3,\frac{1}{2})-\frac{3}{2}\beta (3,1)+\frac{3}{2}\beta (3,\frac{3}{2})-\frac{1}{2}\beta (3,2)$$$$=\frac{2}{5}\cdot \frac{2}{3}\cdot \frac{2}{1}-\frac{1}{2}+\frac{3}{2}\cdot \frac{2}{7}\cdot \frac{2}{5}\cdot \frac{2}{3}\cdot \frac{2}{1}-\frac{1}{4\times 3\times 2\times 1}$$$$=\frac{8}{15}-\frac{1}{2}+\frac{8}{35}-\frac{1}{24}=\frac{37}{168}$$

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